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Infinitely many stationary solutions of discrete vector nonlinear Schrödinger equation with symmetry. (English) Zbl 1191.39011
Authors’ abstract: We study the existence of stationary solutions for the following discrete vector nonlinear Schrödinger equation
$i\frac{\partial\varphi_n}{\partial t}= -\Delta\varphi_n+ \tau_n\varphi_n- Jf(n,|\varphi_n|)\varphi_n,$ where $$\varphi_n$$ is a sequence of 2-component vector, $$J=\left(\begin{smallmatrix}0&1\𝟙&0\end{smallmatrix}\right)$$, $$\Delta\varphi_n= \varphi_{n+1}+ \varphi_{n-1}- 2\varphi_n$$ is the discrete Laplacian in one spatial dimension and sequence $$\tau_n$$ is assumed to be $$N$$-periodic in $$n$$, i.e. $$\tau_{n+N}=\tau_n$$. We prove the existence of infinitely many nontrivial stationary solutions for this system by variational methods. The same method can also be applied to obtain infinitely many breather solutions for single discrete nonlinear Schrödinger equation.

##### MSC:
 39A12 Discrete version of topics in analysis 35Q55 NLS equations (nonlinear Schrödinger equations)
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##### References:
 [1] Bruno, G.; Pankov, A.; Tverdokhleb, Y., On almost-periodic operators in the spaces of sequences, Acta appl. math., 65, 153-167, (2001) · Zbl 0993.39013 [2] Ding, Yanheng, Variational methods for strongly indefinite problems, Interdisciplinary mathematical sciences, vol. 7, (2007), World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ · Zbl 1133.49001 [3] Eilbeck, J.; Johansson, M., () [4] Kryszewski, W.; Szulkin, A., Generalized linking theorem with an application to semilinear Schrödinger equations, Adv. diff. equat., 3, 441-472, (1998) · Zbl 0947.35061 [5] Pankov, A.; Zakharchenko, N., On some discrete variational problems, Acta appl. math., 65, 295-303, (2000) · Zbl 0993.39011 [6] Pankov, A., Gap solitons in periodic discrete nonlinear Schrödinger equations, Nonlinearity, 19, 27-40, (2006) · Zbl 1220.35163 [7] Pankov, A.; Pflüger, K., On semilinear Schrödinger equation with periodic potential, Nonlinear anal., 33, 593-609, (1998) · Zbl 0952.35047 [8] Pankov, A., Gap solitons in periodic discrete nonlinear Schrödinger equations, II: a generalized Nehari manifold approach, Discrete contin. dyn. syst., 19, 419-430, (2007) · Zbl 1220.35164 [9] Li, G.B.; Szulkin, A., An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. contemp. math., 4, 763-776, (2002) · Zbl 1056.35065 [10] Teschl, G., Jacobi operators and completely integrable nonlinear lattices, (2000), American Mathematical Society Providence, RI · Zbl 1056.39029 [11] Ablowitz, M.; Prinari, B.; Trubatch, A., Discrete and continuous nonlinear Schrödinger systems, (2004), Cambridge University Press · Zbl 1057.35058 [12] Weinstein, M., Excitation thresholds for nonlinear localized modes on lattices, Nonlinearity, 12, 673-691, (1999) · Zbl 0984.35147 [13] Guo, Z.M.; Yu, J.S., The existence of periodic and subharmonic solutions to subquadratic second-order difference equations, J. lond. math. soc., 68, 419-430, (2003) · Zbl 1046.39005 [14] Yu, J.S.; Guo, Z.M.; Zou, X.F., Periodic solutions of second order self-adjoint difference equations, J. lond. math. soc., 71, 146-160, (2005) · Zbl 1073.39009 [15] Zhou, Z.; Yu, J.S.; Guo, Z.M., Periodic solutions of higher-dimensional discrete systems, Proc. R. soc. edinb. A, 134, 1013-1022, (2004) · Zbl 1073.39010 [16] Guo, Z.M.; Yu, J.S., Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems, Nonlinear anal., 55, 969-983, (2003) · Zbl 1053.39011 [17] Yu, J.S.; Bin, H.H.; Guo, Z.M., Multiple periodic solutions for discrete Hamiltonian systems, Nonlinear anal. TMA, 66, 1498-1512, (2007) · Zbl 1115.39017 [18] Deng, X.Q.; Cheng, G., Homoclinic orbits for second order discrete Hamiltonian systems with potential changing sign, Acta. appl. math., 103, 301-314, (2008) · Zbl 1153.39012 [19] Ding, Y.H., Multiple homoclinic in a Hamiltonian system with asymptotically or super linear terms, Commun. contemp. math., 8, 453-480, (2006) · Zbl 1104.70013 [20] Ding, Y.H.; Jeanjean, L., Homoclinic orbits for a nonperiodic Hamiltonian system, J. diff. equat., 237, 473-490, (2007) · Zbl 1117.37032 [21] Arioli, G.; Szulkin, A., Homoclinic solutions of Hamiltonian system with symmetry, J. diff. equat., 158, 291-313, (1999) · Zbl 0944.37030 [22] Coti-Zelati, V.; Ekeland, I.; Séré, E., A variational approach to homoclinic orbits in Hamiltonian systems, Math. ann., 288, 133-160, (1990) · Zbl 0731.34050 [23] Ding, Y.H.; Girardi, M., Infinitely many homoclinic orbits of a Hamiltonian system with symmetry, Nonlinear anal., 38, 391-415, (1999) · Zbl 0938.37034 [24] Pankov, A.; Rothos, V., Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity, Proc. R. soc. A, 464, 3219-3236, (2008) · Zbl 1186.35206 [25] Willem, M., Minimax theorems, (1996), Birkhäuser · Zbl 0856.49001
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